Favorite Lesson

I subscribe to several education sites and a popular theme recently has been teachers sharing their favorite lessons.  This got me thinking about whether or not I have a favorite lesson.

A few years ago, it would have been difficult for me to answer this because there would have been several I would have wanted to choose.  “Everything You Need to Know about Logarithms in Less Than 45 Minutes” would have made the short list.  “Crash Day” near the end of the first half of honors pre-calculus when so much material we have covered up to that point goes into proving e^(iπ)+1=0.  Partial fraction decomposition and rotation of conics would have been on the short list as well.  I enjoy the material from these lessons, it’s easy to build a story around them, and the kids were able to follow them (for the most part). 

However, I’m no longer lecturing, and at this point I would have to say that I don’t have a favorite lesson because I don’t have math lessons in my classroom.  Refocusing the course in a Harkness style has meant transferring the class over to a more holistic outlook when it comes to the material.  We don’t focus on individual topics.  Rather, we focus on how the material they already know coming into the course can be extended, utilized, and synthesized.  As such, there aren’t lessons on the individual topics.  Instead, we focus on making a little progress every day with a lot of different topics.  For example, a “normal” day this term could include working with parametric equations and projectile motion, equations of conics, solving trigonometric equations, and solving triangles, all on the same day.  And that’s not an exhaustive list of the material we cover. 

So if pressed to give my favorite lesson, it would have to be the one where I help the kids see that math isn’t a bunch of separate topics, but rather is a unified whole.  In other words, I get to present my favorite lesson every day.

The Goal

I’ve been giving a lot of thought lately to the common core standards and what they are really trying to accomplish.  There are many supporters and many critics, and I find myself somewhere in the middle, mainly because in working with the standards and in trying to prepare for their implementation I have seen the good and the bad.  However, there is one main “theme” that tends to stand out that distinguishes the supporters from the critics, and that is the way the following question is answered: What is the goal of education?

If one sees the goal of education as the transmission of facts or techniques, then the common core standards make no sense whatsoever.  Reading through the standards and looking at some of the workbooks and other materials that have been produced, there is a feeling that the “basic skills” are not being emphasized, whether those basics are knowledge of parts of speech or memorization of arithmetic facts or whatever.  In fact, at a glance, it appears as though these skills are not being valued at all.  As such, the outcry against the common core standards tends to take the form of “just teach them to do the math” or “just teach them to write”.  And in many ways I can understand the complaint. 

However, I believe the complaint comes more from misunderstanding than it does from actual rejection of what the common core is actually trying to do.  You see, the emphasis of the common core is on two things: (1) understand the material well enough to explain it simply; and (2) understand the material well enough to apply it creatively.  This is the goal of education for the common core.  Implied in this is the reality that the students still need to know the basics of writing a sentence and adding fractions.  However, being able to go through the mechanical processes is not enough.  In particular, mechanically getting the correct answer does not show a real understanding of the material.  In many cases, what it shows is the memorization of a process rather than the understanding of a concept, and as such asking a student to explain what they did is important, since in the explanation you will see which of the two (memorization or understanding) the student has actually accomplished.

Unfortunately, the workbook pages I have seen on social media sites complaining about how we need to get rid of the common core seem to be pages from late in the learning process, and therefore are asking the student to go beyond the mechanical process and explain what they did.  The misunderstanding on the part of the critics is that they see these worksheets as trying to teach the students the basics, when what the worksheets are actually trying to do is assess how well the student has understood the material. 

The implications of all of this for the classroom teacher are quite profound.  If all I do is show my students how to perform the standard multiplication algorithm, and then assess them on how well they can perform the algorithm, I will not be checking how well they understand what multiplication actually is nor how well they can use the algorithm in a problem-solving situation.  The same would hold true for teaching a student the parts of speech, or factoring, or many other “basics”.  However, if I emphasize problem-solving in my classroom, and as the need arises show the students the usefulness of a particular algorithm, then the students will still get the basics, but they will do so while developing a deep understanding of the material and creatively applying the basics. 

This is where the discussion-based classroom is at its best.  Students work collaboratively to creatively solve a problem, and in doing so they learn the basics because they need those basics to solve the problem.  This is what the “real world” looks like.  In life, we are rarely confronted with a problem we already know how to solve.  Instead, we are confronted with a situation and we need to go find the tools and develop the skills necessary to solve the problem.  A classroom centered on memorization of the basics does not prepare students for this; a classroom centered on the essential spirit of the common core, i.e., a discussion-based classroom, does, and it does so in a way that does not short-change the basics.

All that being said, the implementation of the testing regimen that is to accompany the common core standards is where this entire process meets with resistance from teachers, even those of us who agree with the intent of the standards.  Asking the kids to be creative and then testing them for conformity is, in my mind anyway, a contradiction, and is a cause for concern.  Yes, some of the exercises are performance-based, but I wonder just how flexible the graders are going to be.  If a kid uses a method that is mathematically valid on an exercise, will the grader have the mathematical prowess to determine the validity of the method?  For that matter, will the rubric constrain the ability of the grader to award credit for a mathematically valid response?    Asking the kids to patiently problem solve and then testing them with a relatively short time constraint is contradictory as well.  Add to these the fact that we are losing several weeks (or more) of regular class time to administer the tests, and you can see where a teacher might be concerned.

So, do I think we need to return to just teaching the basics?  Clearly, the answer is no.  Do I think the current implementation of the standards, and in particular the testing that comes along with them, is in the best interest of the students?  Again, and despite the fact that I agree with the fundamental tenets of the common core, I would have to say no.  There has to be some middle ground, where the basics are included in the bigger picture of problem solving (rather than problem solving being included as an add-on to the teaching of the basics), and where collaborative work and communication skills are the fertile ground in which the creative use of these skills take root and grow.

 

Sounds like Harkness to me.

Baseball Practice

I have been asked a couple times recently why I run my classroom the way I do.  Here is my current response:

It is a beautiful Saturday afternoon and, more importantly, it’s the first day of baseball practice.  The coach, having never met any of the players, quickly introduces himself, and then begins batting practice.  The coach lines up the players behind the backstop while he spends the next hour hitting baseballs being pitched to him by a machine he set up on the mound before practice began.  The boys watch intently as the coach explains and demonstrates how to properly stand, hold the bat, swing the bat, and so on.  At the end of the hour, the coach tells the boys to go home and practice what they have seen.

At practice the following Tuesday evening, the coach asks the players how their practice at home went.  One of the players says that he struggled with his stance, so the coach demonstrates again the proper way to stand in the batter’s box, showing the player by example how it should be done.  Much the same as Saturday, the player watches intently as the coach hits the baseball.

“Do you understand what you were doing wrong?”

“Yep, I think I get it now.”

“Good. Any other questions?”

Silence. 

Then the coach begins fielding practice.  The boys stand behind the backstop and watch intently as the assistant coach hits ball after ball to the head coach who fields ball after ball from the shortstop position.  He shows the players proper technique, how to stand, how to hold and place the glove, and so on, and after an hour of demonstrating what the players are supposed to do, sends them home to practice on their own.

This is repeated over the next several weeks with the coach showing the players how to catch fly balls, how to bunt, how to pitch, how to throw.  He shows them everything they need to know to play baseball. He never sees them play.  They never practice together or play together. The first time the coach actually sees them play baseball is at the first game.  You can imagine the results.

The coach is confused.  He showed them everything they needed to know.  They were very attentive at the practices, and what few questions they had he answered.  How did they not do well?

Absurd as this sounds, this is precisely how many people teach mathematics, performing in front of the class while the students watch, and then sending the students home to practice on their own.  The teacher answers questions by showing the students how to solve the problem at hand.  The teacher never actually sees the students do any mathematics except for the tests.

 

Why do I run my classroom the way I do?  Much the same as a coach spends most of his time watching the players at practice, noting the mistakes they are making and helping them correct those mistakes before the actual game rolls around, I want to see my kids doing math, see the mistakes they are making, and help them correct those mistakes as they are learning the material.  Just as the majority of the time spent at baseball practice involves the players playing baseball or practicing a particular skill, I want the majority of the time in class to involve the kids actually doing math.

Why it took me so long to figure this out is a travesty.  How others don’t see the benefits of running a student-centered classroom is a mystery.

Differentiating

At the end of the last unit, quite a few of the students were struggling with the basics of vectors, as the grades on the last test made evident.  Knowing that some of the material to follow depended on understanding this information, the other honors pre-calculus teacher and I came up with an optional replacement assignment for the students to do.  The descrpition itself was simple: convince me you know what you're doing when it comes  to addition, subtraction, scalar multiplication, finding unit vectors, determining component form from magnitude and direction, and determining magnitude and direction from component form.  How this was to be accomplished we didn't specify.  They could write a short paper, make a poster, make a video...whatever.  The "how" wasn't important.  Just convince me you know what you're doing, and the score you earn will replace the score for the vector exercise on the test.

To say that it went well would be an understatement.  Granted, overall what the kids turned in wasn't necessarily creative, as most turned in explanations of the above topics with an example of each (though one student did write new lyrics to "The Devil Went Down to Georgia" and performed the song in class), but afterwards the students said that they definitely learned a lot from doing the activity, and the subsequent exercises we have done in class have convinced me that this is, in fact, the case.

So, after the most recent test, we have given the students the opportunity to choose one problem from the test and turn in a similar project to convince us they really do understand the material the particular exercise was assessing.  Again, we were specific enough so they know what skill/knowledge we want them to demonstrate, but beyond that the assignment is fairly open-ended.  And I have no doubt that when they turn in the assignment next Wednesday, we will have achieved the same result as before: quality work from which the students learn the material well.

Notice by the way that none of this is focused on "extra credit".  The focus is on understanding the material and demonstrating that understanding, period.  Yes, the assignment helped the kids' grades, and while that may well have been the initial motivation for doing the assignment, I'm hopeful that enough of them got enough out of the first assignment that they will see the true importance of doing the second one.

All of this got me wondering: Why do we have the students complete some sort of in-class assessment wherein the students have to demonstrate their mastery of the material by doing exercises prescribed by the teacher?  If the answer is: because that's how they're going to be assessed on state-mandated standardized tests, or because that's how they're going to be assessed on college entrance exams, or that's the way they're going to be assessed in the next course or in college, or because we've always done it this way, then haven't we missed the point of assessing the students?  These may well appear to be practical reasons, but are they valid?  I understand that there needs to be some sort of assurance that the students did their own work, and for the record we achieved that with the assignment described above since the exercises that have been assigned since then that depended on knowledge of that material have been completed well by the students.  And there are plenty of teachers I know who, for various reasons, would never dream of giving an assignment as open-ended as the one described above.  But seriously, if the point of assessing the students is to give them an opportunity to demonstrate their mastery of the material, then how is the above assignment any different than a short essay in an English or history class?

My current conclusion: we need to give the students more variety when it comes to the opportunities we give them to demonstrate their understanding of the material.  There needs to be accountability, but we need to admit that, in the same way that we differentiate instruction, we need to differentiate assessments.

Creativity

So here's a question from the exercises that were assigned during the past week:

You have often wondered how tall the water tower near Pine Hill Park is, so one day you take your protractor and head for the park.  Looking at the top of the tower, you measure the angle of elevation to be 30°.  After moving 108 feet closer to the tower, you again measure the angle of elevation to the top of the tower and find it to be 50°.  What is the height of the tower?

A pretty standard exercise, I admit.  Here's the thing: the kids came up with at least four different ways to determine the height of the tower based on this information.  Some used a system of equations involving tan(30) and tan(50).  Some used the law of sines.  Some used the fact that the "large" triangle is a 30-60-90 triangle.  Some used the Pythagorean Theorem.  Many used some combination of the above, but not necessarily the same combination.  Most arrived at the correct answer.

Among others, this is one of the things I love best about teaching through discovery and discussion.  The kids came into the class knowing basic right-triangle trig, and they had used special right triangles before.  Through the course of the exercises, we have emphasized both of these topics, along with discovering the law of sines.  That sets us up to work on this exercise.  If I had done a similar problem in class as an example to "show them how" the problem is to be done, the next day I would have had classes full of kids who all did this problem exactly the same way I did it.  No critical thinking, no real problem solving, no creativity, just mindless imitation.  Instead, the kids had to make sense of the exercise themselves, use the "tools" available to them, and solve the exercise.  Working through things this way sets them up to handle a wide variety of similar exercises that don't necessarily ask them for the height of the tower, but that could instead ask them for the distance from the base of the tower, the angle of elevation, etc., whereas me giving them as example and having them imitate it sets them up to do an identical exercise with the different numbers and nothing more.  To ask them to find the angle of elevation would require a separate example, and asking them to find the distance from the tower would require a third, none of which build or even encourages critical thinking, problem solving, or creativity.  Instead, it promotes memorization, and little to nothing else...memorization which, by the way, fades away quickly.  And if what the kids memorize is inaccurate, then fixing it is difficult.  I have found that it is much easier to correct a reasonable but flawed guess about which a kid isn't completely sure than it is to correct a flawed bit of memorization about which a kid is absolutely certain.

Sadly, I've seen teachers essentially train kids to do exercises only and exactly the way that they are shown in class and then get upset that the kids aren't able to generalize the process on the test.  Worse, they don't make the connection that if we don't regularly ask the kids to understand the material well enough that they can explain it and be creative with it, then they won't be able to put the pieces together and generalize the process once every 2-3 weeks on a test.  Of course, rock bottom is the teachers whose test questions are only ever the same exercises as those that were specifically shown as examples in class, just with different numbers.  This kind of teaching may have cut it in the past, but it won't now, especially with end-of-course exams that require creativity and critical thinking coming soon to our classrooms.

If we want the kids to be creative, problem-solving, critical thinkers, then we need to ask them to be creative, problem-solving, critical thinkers every day.  We need to encourage them to make a conjecture as to how a problem can be solved, attempt to solve the problem based on that conjecture, and right or wrong learn from the experience.  This requires us to create an atmosphere in our classrooms where it's safe to make a mistake and then try something else.  It requires us to see things the way they do and help them find a way that the material makes sense to them.  It requires us to accept different but mathematically valid methods of solving an exercise.  

It's easier to just ask the kids to do things the way we do them.  It's easier to grade a set of papers that all look the same.  However, good teaching doesn't focus on the teacher.  Good teaching focuses on the learning being done by the students.  It isn't easy.  But if we don't want the kids to be creative, problem-solving, critical thinkers because it makes things difficult for us, then we need to get out of the classroom.

Physical Space

During the last week, as we prepared for the second test of the trimester, it occurred to me just how profoundly the physical space in which a class is held can influence the way in which the class is run.  I am fortunate enough to have tables that seat two students apiece as well as an entire long wall (20+ feet) on which is a white board complete with sliding panels.  As the room changed this week from the usual “groups” configuration to “stadium” style for the review to the more traditional rows for the test, I was first grateful to have the ability to change the room so easily for all of these different needs.  However, it got me thinking about what I would do differently if I were to design the room “from the ground up”.

First, I would include more board space.  Yes, I know I already have more board space than most.  However, to comfortably run four groups of 7-8 students, I would need at least one more wall worth of white boards.  This is probably more indigenous to a math class, but since that’s what I teach, it’s one thing that I would definitely include.

Second, I would get rid of the rectangular tables and replace them with something that would allow for better eye contact and give more of a community feel to each group.  I've seen large trapezoid-shaped tables and I think that’s probably what I would opt for, but I’m not certain.  A little more research would be needed before making a final decision.  I would also equip each of the groups of tables with a tablet computer for looking up any information they may need during class.

Third, I would get rid of my teacher desk and replace it with stand-up desk that can accommodate a desktop computer (or maybe a laptop/tablet combo), along with a tall chair.  I rarely sit at my desk during the day, and quite honestly it takes up valuable space that could be used to give the kids a little more room.  However, during class I am constantly going back-and-forth to my lectern to make notes about what I’m observing and the feedback I am getting from the kids about how well they understand the material.  A stand-up desk would make a lot more sense for me.

Now don’t get me wrong.  I am acutely aware that I already have far more in my classroom than most, and I’m extremely grateful for it all.  But, if we’re really going to ask the kids to do more collaborative work in class, then we need to be aware of the needs we have of the physical space itself to make it happen, and in most instances, the traditional classroom with one white board and 30 individual student desks just won’t cut it.

Feedback

As the year has progressed, and the more comfortable I have become over the last two years with using discussions in my classroom, I have struggled with trying to keep track of everything.  I have become firmly convinced that the most important thing I do in the classroom is pay attention to the kids: find out what they know, what they don't, and determine what is preventing them from making the progress they need to make.  Admittedly, the opportunity to do all of this is a huge advantage of running a Harkness classroom (or a "flipped" classroom, for that matter) as opposed to a "traditional" one.  In the past, the amount of information I was able to get from the kids on a daily basis was extremely limited, mainly because I was the one doing most of the talking.  Now, I have the opportunity to get a good feel for where each and every one of the kids is every day.  However, trying to write down notes in enough detail that I can remember and then act on what I am observing has proven to be a huge undertaking.

My solution so far has been to make a spreadsheet on my tablet and carry it with me from table to table.  The spreadsheet has the names of the students down the first column, and the skills they are to be learning during the course of the trimester across the top row.  During class, I mark an "x" in the appropriate cell when one of the kids presents a solution on the board or when they make a solid summary statement about one of the concepts to the group.  After each test, I go through the spreadsheet and mark a "t" in the appropriate cell if the student earned at least a "B" on the exercise that was on the test that relates to a particular skill.  All of this has been very helpful, and some patterns are certainly emerging in terms of who goes to the board to do only review exercises, who rarely goes to the board at all, and who has a reasonable mastery of all of the skills to the point that they are willing  to go to the board at any time.  It's not perfect, because I still feel I miss a lot of what happens at the other tables when I get deeply involved in the discussion at one table, but it's certainly better than what I was able to do in the past, when most if not all of the feedback I got from the kids came in the form of a formal quiz or test.

This leads me to one of the other big revelations I've had recently.  There has been a lot of talk at our school recently on the topic of "feedback", mostly in the form of discussing the work of John Hattie.  Initially, my reaction was, "OK. how can I give the kids more information about what they're doing right, about where they still need some extra practice,..." etc.  However, it occurred to me that the only way for me to give feedback to the kids is if I'm getting feedback from them.  Trying to give them feedback every day requires that I get feedback from them every day.  This kind of interaction, the give-and-take in both directions necessary to allow the two-way feedback happen, is precisely what was missing from my classroom in the past.  This interaction is not only promoted by a discussion-based classroom, it is the very heart of it.  Whether the classroom is running on a Harkness model (which I would classify as discovery-based discussion), a true Direct Instruction model (brief lecture followed by lots of closely monitored in-class practice in the form of exercises or activities done in small groups), or a flipped model (somewhere between Harkness and DI), the important part of the time in class is when the students and teacher are getting and giving feedback to one another.

In short, the most important part of any classroom should be the real interaction between the teacher and the students.  Discoveries can be made anywhere, and lectures can be delivered by online videos, but the frequent feedback necessary for the teacher to know where the students are in their learning and for the students to receive affirmation of or correction to their progress and conclusions can only happen during real dialogue, and this happens best in a discussion-based classroom.

Is it easy to keep up with everything?  No, but it's certainly worth the effort.  And I'll definitely take this over the only-way dissemination of knowledge that used to take place in my classroom.